Transcript 5.1

Section 5.1
Inverse Functions
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives

Determine whether a function is one-to-one, and if it is,
find a formula for its inverse.
 Simplify expressions of the type f f 1  x 
1
f
f  x .
and




Inverses
When we go from an output of a function back to its input
or inputs, we get an inverse relation. When that relation is
a function, we have an inverse function.
Interchanging the first and second coordinates of each
ordered pair in a relation produces the inverse relation.
Consider the relation h given as follows:
h = {(8, 5), (4, 2), (–7, 1), (3.8, 6.2)}.
The inverse of the relation h is given as follows:
{(5, 8), (–2, 4), (1, –7), (6.2, 3.8)}.
Inverse Relation
Interchanging the first and second coordinates of each
ordered pair in a relation produces the inverse relation.
Example
Consider the relation g given by g = {(2, 4), (–1, 3), (2, 0)}.
Graph the relation in blue. Find the inverse and graph it in
red.
Solution: The relation g is
shown in blue. The inverse
of the relation is
{(4, 2), (3, –1), (0, 2)}
and is shown in red. The
pairs in the inverse are
reflections of the pairs in g
across the line y = x.
Inverse Relation
If a relation is defined by an equation, interchanging the
variables produces an equation of the inverse relation.
Example
Find an equation for the inverse of the relation:
y = x2  2x.
We interchange x and y and obtain an equation of
the inverse:
x = y2  2y.
Graphs of a relation and its inverse are always
reflections of each other across the line y = x.
Graphs of a Relation and Its Inverse
If a relation is given by an equation, then the solutions of
the inverse can be found from those of the original
equation by interchanging the first and second
coordinates of each ordered pair. Thus the graphs of a
relation and its inverse are
always reflections of each other
across the line y = x.
One-to-One Functions
A function f is one-to-one if different inputs have
different outputs – that is,
if
a  b,
then
f (a)  f (b).
Or a function f is one-to-one if when the outputs are
the same, the inputs are the same – that is,
if
f (a) = f (b),
then
a = b.
Inverses of Functions
If the inverse of a function f is also a function, it is
named f 1 and read “f-inverse.”
The –1 in f 1 is not an exponent.
f 1 does not mean the reciprocal of f and f 1(x) can
1
.
not be equal to
f (x)
One-to-One Functions and Inverses
If a function f is one-to-one, then its inverse f 1 is a
function.
• The domain of a one-to-one function f is the range of
the inverse f 1.
• The range of a one-to-one function f is the domain of
the inverse f 1.
• A function that is increasing over its domain or is
decreasing over its domain is a one-to-one function.
•
Horizontal-Line Test
If it is possible for a horizontal line to intersect the graph
of a function more than once, then the function is not
one-to-one and its inverse is not a function.
not a one-to-one function
inverse is not a function
Example
From the graph shown, determine whether each function
is one-to-one and thus has an inverse that is a function.
No horizontal line intersects
more than once: is one-to-one;
inverse is a function
Horizontal lines intersect more
than once: not one-to-one;
inverse is not a function
Example
From the graph shown, determine whether each function
is one-to-one and thus has an inverse that is a function.
No horizontal line intersects
more than once: is one-to-one;
inverse is a function
Horizontal lines intersect more
than once: not one-to-one;
inverse is not a function
Obtaining a Formula for an Inverse
If a function f is one-to-one, a formula for its inverse
can generally be found as follows:
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve for y.
4. Replace y with f 1(x).
Example
Determine whether the function f (x) = 2x  3 is oneto-one, and if it is, find a formula for f 1(x).
Solution: The graph is
that of a line and passes
the horizontal-line test.
Thus it is one-to-one and
its inverse is a function.
1. Replace f (x) with y:
y = 2x  3
2. Interchange x and y:
x = 2y  3
3. Solve for y:
x + 2 = 3y
x3
y
2
4. Replace y with f 1(x):
f
1
x3
x  
2
Example (cont)
x3
Graph f x   2x  3 and f x  
2
using the same set of axes. Then compare the two
graphs. The solutions of the inverse function can be
found from those of the original function by
interchanging the first and second coordinates of each
ordered pair.
1
Example (continued)
The graph f 1 is a reflection of the graph f across the
line y = x.
Inverse Functions and Composition
If a function f is one-to-one, then f 1 is the unique
function such that each of the following holds:

f 1 o f
x  
f 1  f x   x,
f o f x   f f x  x,
1
1
for each x in the
domain of f, and
for each x in the
domain of f 1.
Example
Given that f (x) = 5x + 8, use composition of functions to
x8
1
show that
f x  
.
5

f 1 o f
x  
f 1  f x 
 f 1 5x  8 
5x  8   8


5
5x

x
5




f o f 1 x   f f 1 x 
 x  8
 f
 5 
 x  8
 5
8

 5 
 x88  x
Restricting a Domain
When the inverse of a function is not a function, the
domain of the function can be restricted to allow the
inverse to be a function. In such cases, it is convenient to
consider “part” of the function by restricting the domain of
f (x).
Suppose we try to find a formula for the inverse of
f (x) = x2.
y  x2
xy
2
 xy
This is not the equation of a
function because an input of 4
would yield two outputs, 2 and 2.
Restricting a Domain
However, if we restrict the domain of f (x) = x2 to
nonnegative numbers, then its inverse is a function.
If f x   x 2 , x  0, then f 1 x  
x.